Mechanical Testing and DiagnosisISSN 2247 – 9635, 2012 (II), Volume 4, 70-78

ANALYSIS OF NONCIRCULARGEARS MESHING

Marius VASIE, Laurenţia ANDREI

”Dunarea de Jos” University of Galati, ROMANIA

v_marius_gl@yahoo.com

ABSTRACTBased on a general procedure, that uses the supershape equation to

generate the pitch curve of the driving gear, and on an analytical approachfor the generation of tooth profiles, the paper presents an analysis of thenoncircular gear meshing. A first step considers the simulation of rolling, in2D space, and validates the geometrical and kinematical calculations madein the design phase. Further, the 3D path of contact is analyzed and itsdistribution and evolution during gears meshing are monitored. Knowinghow the path of contact evolves during gear meshing is important for furtherinvestigations such as stress and displacement distributions. The lack ofresearches regarding the meshing conditions of noncircular gears highlightsthe importance of this paper.

Keywords: noncircular gears, meshing analysis, path of contact

1. INTRODUCTION

Noncircular gears are presented as a curiosity for the gear industry history, due totheir complex design and manufacturing difficulties. Nowadays, performant modelling andsimulation softwares, advanced CNC machine tools and nonconventional manufacturingtechnologies enable noncircular gear design and manufacture to be more feasible and tocontinuously chalenge the scientist. As mechanisms used to generate variable motion laws,in comparison with cams, linkages, variable transmission belts, Geneva mechanisms andeven electrical servomotors, noncircular gears are remarkable due to their advantages, suchas the ability to produce variable speed movements in a simple, compact and reliable way,the lack of gross separation or decoupling between elements, fewer parts in the designphase, the ability to produce high strength-to-weight ratios, etc [1]. The use of noncirculargears in industry certifies their performances, leading to new ideas for improved workingconditions.

Generation of noncircular gear is usually developed starting from the hypothesis suchas the law of driven gear motion, variation of gear transmission ratio, design of driving gearpitch curve [2-4]. Based on the gear pitch curves, the tooth representation methods considerthe theory of envelope surfaces [5], analytical approaches [6] or simulation of the cuttingprocess [7]. Virtual modelling of the noncircular gears, essential for manufacture, could

Mechanical Testing and Diagnosis, ISSN 2247 – 9635, 2012 (II), Volume 4, 70-7871

also be used for further development of a vast theoretical research on meshing conditions,with objectives like path of contact analysis, stress and displacement analysis, the influenceof teeth’s geometrical parameters and generation and positioning errors of the gears on thegear meshing, etc. From this point of view, the literature is lacunar compared to standardgears, the lack of information being accounted for, perhaps, by the wide range of shapesand materials, lack of standard procedures, etc.

Following the traditional algorithm for noncircular gear generation, the paper brieflypresents a general method for the design of gears whose geometry is defined by a constantpressure angle. The main objective of the paper is to investigate the meshing conditions ofthe designed noncircular gears, highlithing the path of contact evolution and distributionalong the teeth placed in the vicinity of the pitch curve minimum radius, where interferencecould occur and maximum stresses are induced.

2. NONCIRCULAR PITCH CURVES GENERATION

The first step in the noncircular gears virtual design process is the generation of theconjugate pitch curves, starting from a predesigned law of motion for the driven element ora predesigned geometry for the driving gear pitch curve. Based on the second mentionedhypothesis, the authors generalise the driving gear pitch curve by introducing the Gielis’supershape [8], defined by:

132 n

1n

1n

111 4

nsinb1

4ncos

a1r

(1)

where a, b are non-zero real numbers that define the semi-lengths of the classical ellipse; n- a real number that multiplies the polar angle and defines the number of lobes of thesupershape, ie its rotational simmetry; n1, n2 and n3 - real non-zero numbers that lead topinched, bloated or polygonal, symmetric or asymetric shapes, depending on their valuesand relation.

Table 1 illustrates several families of supershapes, scaled to corespond in lengthwith a standard gear pitch curve defined by modulus m = 2 mm and number of teeth z = 24.The exponents n2, n3 are considered equal and their variation is chosen as n2 = n3 [2, 10](the continuity imposed for the second derivative of the defining function of thesupershape, in order to be used in gear theory, requires that exponents n2, n3 2). Otherdefining parameters are mentioned in Table 1. It can be noticed that for n2 = n2 = 2, circularpitch curves are obtained; odd number of lobes and different semiaxis lengths lead to opencurves; as the exponent n1 is increased, pitch curve exibits concave zones and the minimumcurvature radius decreases. An extremely versatile curve, the supershape is quitespectacular but it is obvious that, in order to obtain appropriate gear pitch curves, with nopinched or no self-intersecting shapes, with curvatures that exclude the further risk ofundercutting during teeth generation, a proper selection of the defining parameters must bedone [9].

Once the potential pitch curve geometry is chosen, the next important steps arefocused on the determination of the gears center distance and the conjugate pitch curvegeometry, respectively. Due to complex equation of the supershape, the determination ofthe gear center distance, D, is made through numerical integration and iterative procedure,in Matlab environment, and it is based on the assumption that the driving pitch curveperforms N1 revolutions for one revolution of the driven pitch curve and its length is scaledproportionally, to correspond to gear modulus m and number of teeth z1.

Closed pitch curves for the driven gears, defined by z2 number of teeth, are generatedif [10]:

Mechanical Testing and Diagnosis, ISSN 2247 – 9635, 2012 (II), Volume 4, 70-7872

Table 1. Families of supershapes

a = b = 1, n = 1, n1 = 0,5 a = b = 1, n = 2, n1 = 1

a = 1.5, b = 1, n = 3, n1 = 3 a = 1.5, b = 1, n = 3, n1 = 2

1

2

0 11

111

2

0 22

111

2

0 1121d

rDrd

rrd

m1

N2

(2)

where m12 is the gear ratio and r2, θ2 – the polar coordinate of the driven pitch curve.The iterative procedure is controlled by an accuracy established at:

61

2

0 11

11

110d

rDr

N2

(3)

Once the gear center distance is obtained, the driven pitch curve can be represented,defined by the following equations:

11122 rDr (4)

1 1

01

0 11

111

1212 d

rDrd

m1

(5)

Figure 1 illustrates examples of conjugated noncircular pitch curves, defined by thefollowing common design parameters: the supershape semiaxis lengths a = 1.5, b = 1, thegear modulus m = 2 mm and the number of revolution of the driving pitch curve N1 = 1.Other specific defined parameters are mentioned in the tabel.

Mechanical Testing and Diagnosis, ISSN 2247 – 9635, 2012 (II), Volume 4, 70-7873

a) b)Fig. 1. Examples of conjugated noncircular pitch curves

a) convex pitch curves (n = 4, n1 = 4, n2 = n3 = 3, z1 = 48)b) convex-concave pitch curve (n = 6, n1 = 4, n2 = n3 = 2.5, z1 = 60)

a) b)

Fig. 2. Tooth profiles of noncircular gears with convex (a) and convex-concave (b) pitch curves

3. TOOTH PROFILES GENERATION

Literature review highlights several methods for noncircular gears’ tooth profilesgeneration, based on analytical procedure, enveloping theory and manufacturingsimulation. Considering analytical generation of the gear tooth, the complexity of the pitchcurve’s defining equation (Eq. 1) requires the development of specific Matlab codes toenable mathematical calculations with a high accuracy.

The analytical generation of the gear tooth flanks, proposed by the authors [11], isbased on the rolling movement of one standard rack cutter’s tooth over the pitch curve ofthe noncircular gear. The positioning and rolling of the tool at the level of each tooth satisfythe geometrical and kinematical conditions of gear – rack engagement. After the toothflanks generation, in Matlab environment, the planar curves are imported, throughAutoLISP codes, and edited in AutoCAD, in order to represent the final solid gear model.Fig. 2 illustrates tooth profiles of the gears defined by the pitch curves presented in Fig. 1,with details subtracted from both Matlab and AutoCAD graphic screens.

4. ANALYSIS OF NONCIRCULAR GEARS’ MESHING

The analysis of the noncircular gear’s meshing, in virtual environment, is based on theaccuracy and precision of the geometrical and kinematical calculations made in the phase

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of tooth profiles generation and aims the engagement simulation and evolution of the gearspath of contact.

4.1. Gears’ 2D Meshing

An initial check out of the gear tooth profiles generation is developed by the 2Dsimulation of gears meshing, in AutoCAD environment. Thus, the rotational angle for thedriven gear (Eq. 5), calculated through numerical integration in Matlab, is imported toAutoCAD and transmited to the driven gear while the driving gear perform one revolution.Table 2 captures particular positions of the engagement of one of the two pairs ofnoncircular gears previously presented (Fig. 2), defined by the following parameters: a =1.5, b = 1, n = 4, n1 = 4, n2 = n3 = 3, m = 2 mm, z1 = 48, N1 = N2 = 1. Zooming the meshingzone, no interference could be noticed, even in sensitive zones with smaller curvatureradius, certifying that the gear design procedures were properly developed.

Table 2. Noncircular gears’ engagement

θ1 = 20.01o, θ2 = 25.40o θ1 = 45.03o, θ2 = 54.41o

θ1 = 60.04o, θ2 = 67.89o θ1 = 90.06o, θ2 = 89.97o

4.2. Gears Path of Contact

The path of contact analysis, performed with solid models of noncircular gears, isnecessary for purposes such as: qualitative and quantitave information on teeth engagementand for further studies on the state of stress and displacements. The gears 3D meshingsimulation points out the distribution and evolution of the contact between the conjugatedtooth flanks. The study is carried out according to the following algorithm:

Starting by defining parameters of the supershape, for a chosen driving geargeometry and kinematics, the tooth profiles of the noncircular gears train are generated inMatlab environment and imported to AutoCAD, where the gear solid models are generated;

On the driving gear, the position of the gear tooth corresponding to the minimumpitch curve radius is chosen to be analyzed (Fig. 3). For this tooth, the angular variationdefined by the position of engagement entry and exit points, respectively, is extracted fromMatlab results. Six rotational angles are selected, within this variation domain, where the

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path of contact is highlighted and investigated, in terms of tooth distribution andcorresponding area. It should be specified that the path of contact is figured for a controlledinterference, obtained through an additional rotational motion of the driving gear.

Fig. 3. 3D models of noncircular gears

Tabel 3. Path of contact for noncircular gears with three lobes

θ1 = 52.07o, θ2 = 53.95o θ1 = 54.57o, θ2 = 55.88o

θ1 = 57.08o, θ2 = 57.78o θ1 = 60.08o, θ2 = 60.06o

θ1 = 62.58o, θ2 = 61.96o θ1 = 65.09o, θ2 = 63.86o

Table 3 presents the evolution of the contact path of the noncircular gear presented inFig. 3. Tooth number zi = 12 corresponds to the minimum radius of the pitch curve, it hasthe active flank defined by the angular interval θ1 [52.07o, 65.09o] and receives an

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additional rotation of 0.011o. It was found that there are always two teeth in contact. Thetooth starts the engagement with an area of the contact pattern of 9.34 mm2, reaches themaximum value of 14.87 mm2, in the fifth rotational position, after a rotation of 62.58o forthe pinion and 61.96o for the driven gear, leaving the engagement with a minimum value ofthe contact pattern of 5.70 mm2.

4.3. Influence of Supershape Defining Parameters on Gear Path of Contact

It is obvious that each supershape defining parameter, influencing the gear pitch curvegeometry, will influence the caractheristics of the gear path of contact. A brief study of thenoncircular gear meshing, of the design hypothesis above mentioned by the authors, isfuther developed, the main objective being the proper choice of the potential parameter thatwould lead to an enhanced gear power transmission capacity.

Figure 4 illustrates the influence the pitch curve number of lobes (n) has on the area ofthe path of contact monitorized on the tooth placed on the minimum curvature radius. Thegear train is defined by a = b = 1, n1 = 1, n2 = n3 = 3, m = 2, z = 54, N1 = N2 = 1 and thecontroled interference is 0.015o. It could be noticed that:

Fig. 4. Evolution of the gear path of contact for gears with one, two and three lobes

- for n = 1, the tooth gets into meshing after the pinion is rotated at – 9.0251o (rotationin clockwise direction), recording an area of the path of contact of 8.3335 mm2, gets themaximum surface contact (16.6859 mm2), at θ1 = -7.0195o, and leaves the meshing at θ1 =5.0139o, with a contact of 6.9412 mm2;

- for n = 2, at starting meshing point, the contact path area is recorded 3.3589 mm2,reaches the maximum area of 13.6959 mm2 and decreases to 2.3192 mm2 when the toothflank is about to leave the meshing. The gears contact occurs during the pinion rotationalangle θ1 [-3.0083o, 6.6221];

- for n = 3, the tooth contact increases from the starting area of 2.6458 mm2 tomaximum 13.6056 mm2 and decreases to 2.5945 mm2 at contact exit points.

So, as the number of lobes of the driving gear pitch curve is increased, the tooth isengaged along a shorter rotational angle and the area of the contact path is decreased.

Figure 5 illustrates the evolution of the gear path of contact for noncircular gears thatvaries in pitch curve geometry due to the main exponents n1 (the designed parameters are: a

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= b = 1, n = 4, n2 = n3 = 3, m = 2 şi z = 54). For the same defined interference of 0.015, itis noticed that an increase in n1, even if it reduces the concavity of the pitch curves, doesnot significaly influence the area of the tooth path of contact, and sligthly reduces thecontact rotational angle.

Fig. 5. Influence of exponent n1 on the gear path of contact

Fig. 6. Influence of exponent n3 on the gear path of contact(a = b = 1, n = 2, n1 = 2, n2 = 4, m = 2, z = 54)

When one of the exponents of the trigonometrical terms in supershape equation arevaried (Fig. 6), the supershape exhibits important changes in geometry, relative toconvexity and symmetry. Therefore, the tooth path of contact follows a peculiar behaviourthat could be explained strictly in accordance to the gear geometry.

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5. CONCLUSIONS

Most papers published so far, dealing with noncircular gears, are focused on pitchcurve and tooth profiles modeling methods or on finding applications for these gears. Fewresearchers have analyzed and communicated results on the performance of these gears.Taking advantage of the processing and analysis capacity of the latest generation softwares,the authors have conducted further investigations on noncircular gears’ behavior, designedin the hypothesis of the driving pitch curve generation, as supershape, with constantpressure angle.

The visualization of the engagement, in both 2D and 3D spaces, allowed thevalidation of the engagement’s accuracy, respectively of the geometrical and kinematicalcalculations specific to the phases of gear generation process. The three-dimensionalinvestigation of the solid noncircular gears meshing generated important dates as regard tothe distribution and evolution of the gear path of contact, useful for a better design of thegear geometry and further investigation on noncircular gear behaviour, such as distributionof loads, maximum stress and displacement etc.

ACKNOWLEDGEMENTS

The work of Marius Vasie was supported by Project SOP HRD – EFICIENT61445/2009.

REFERENCES

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